3.3 Reheating after inflation

We discuss the dynamics of reheating and the resulting particle production in the Jordan
frame for the model (3.6). The inflationary period is followed by a reheating phase in which the
second derivative can no longer be neglected in Eq. (3.8). Introducing , we
have

Since during reheating, the solution to Eq. (3.28) is given by that of the harmonic
oscillator with a frequency . Hence the Ricci scalar exhibits a damped oscillation around :

Let us estimate the evolution of the Hubble parameter and the scale factor during reheating in more
detail. If we neglect the r.h.s. of Eq. (3.7), we get the solution . Setting
to derive the solution of Eq. (3.7), we obtain [426]

where is the time at the onset of reheating. The constant is determined by matching Eq. (3.30)
with the slow-roll inflationary solution at . Then we get and

Taking the time average of oscillations in the regime , it follows that .
This corresponds to the cosmic evolution during the matter-dominated epoch, i.e., . The
gravitational effect of coherent oscillations of scalarons with mass is similar to that of a
pressureless perfect fluid. During reheating the Ricci scalar is approximately given by , i.e.

In the regime this behaves as

In order to study particle production during reheating, we consider a scalar field with mass .
We also introduce a nonminimal coupling between the field and the Ricci scalar [88].
Then the action is given by

where . Taking the variation of this action with respect to gives

We decompose the quantum field in terms of the Heisenberg representation:

where and are annihilation and creation operators, respectively. The field can be quantized in
curved spacetime by generalizing the basic formalism of quantum field theory in the flat spacetime. See the
book [88] for the detail of quantum field theory in curved spacetime. Then each Fourier mode obeys
the following equation of motion

where is a comoving wavenumber. Introducing a new field and conformal time
, we obtain

where the conformal coupling correspond to . This result states that, even though (that
is, the field is minimally coupled to gravity), still gives a contribution to the effective mass of . In
the following we first review the reheating scenario based on a minimally coupled massless field (
and ). This corresponds to the gravitational particle production in the perturbative
regime [565, 606, 426]. We then study the case in which the nonminimal coupling is larger than the
order of 1. In this case the non-adiabatic particle production preheating[584, 353, 538, 354] can occur via
parametric resonance.

3.3.1 Case: and

In this case there is no explicit coupling among the fields and . Hence the particles are
produced only gravitationally. In fact, Eq. (3.38) reduces to

where . Since is of the order of , one has for the mode deep inside the
Hubble radius. Initially we choose the field in the vacuum state with the positive-frequency
solution [88]: . The presence of the time-dependent term leads to
the creation of the particle . We can write the solution of Eq. (3.39) iteratively, as [626]

After the universe enters the radiation-dominated epoch, the term becomes small so that the
flat-space solution is recovered. The choice of decomposition of into and is not
unique. In curved spacetime it is possible to choose another decomposition in term of new
ladder operators and , which can be written in terms of and , such as
. Provided that , even though , we have .
Hence the vacuum in one basis is not the vacuum in the new basis, and according to the new
basis, the particles are created. The Bogoliubov coefficient describing the particle production is

The typical wavenumber in the -coordinate is given by , whereas in the -coordinate it
is . Then the energy density per unit comoving volume in the -coordinate is [426]

where in the last equality we have used the fact that the term approaches 0 in the early and late
times.

During the oscillating phase of the Ricci scalar the time-dependence of is given by
, where and ( is a constant). When we
evaluate the term in Eq. (3.42), the time-dependence of can be neglected.
Differentiating Eq. (3.42) in terms of and taking the limit , it follows that

where we used the relation . Shifting the phase of the oscillating factor by
, we obtain

The proper energy density of the field is given by . Taking into account
relativistic degrees of freedom, the total radiation density is

which obeys the following equation

Comparing this with the continuity equation (2.17) we obtain the pressure of the created particles, as

Now the dynamical equations are given by Eqs. (2.15) and (2.16) with the energy density (3.45) and the
pressure (3.47).

In the regime the evolution of the scale factor is given by , and
hence

where we have neglected the backreaction of created particles. Meanwhile the integration of Eq. (3.45) gives

where we have used the averaged relation [which comes from Eq. (3.33)]. The
energy density evolves slowly compared to and finally it becomes a dominant contribution to
the total energy density () at the time . In [426] it was found
that the transition from the oscillating phase to the radiation-dominated epoch occurs slower compared to
the estimation given above. Since the epoch of the transient matter-dominated era is about one order of
magnitude longer than the analytic estimation [426], we take the value to
estimate the reheating temperature . Since the particle energy density is converted to
the radiation energy density , the reheating temperature can be estimated
as4

As we will see in Section 7, the WMAP normalization of the CMB temperature anisotropies determines the
mass scale to be . Taking the value , we have . For
the universe enters the radiation-dominated epoch characterized by , , and
.

3.3.2 Case:

If is larger than the order of unity, one can expect the explosive particle production called preheating
prior to the perturbative regime discussed above. Originally the dynamics of such gravitational preheating
was studied in [70, 592] for a massive chaotic inflation model in Einstein gravity. Later this was extended
to the f (R) model (3.6) [591].

As long as is larger than the order of unity, the last two terms in the bracket of Eq. (3.51) can be
neglected relative to . Since the Ricci scalar is given by Eq. (3.33) in the regime , it
follows that

The oscillating term gives rise to parametric amplification of the particle . In order to see this we
introduce the variable defined by , where the plus and minus signs correspond
to the cases and respectively. Then Eq. (3.52) reduces to the Mathieu equation

where

The strength of parametric resonance depends on the parameters and . This can be described by a
stability-instability chart of the Mathieu equation [419, 353, 591]. In the Minkowski spacetime the
parameters and are constant. If and are in an instability band, then the perturbation
grows exponentially with a growth index , i.e., . In the regime the resonance
occurs only in narrow bands around , where , with the maximum growth index
[353]. Meanwhile, for large , a broad resonance can occur for a wide range of
parameter space and momentum modes [354].

In the expanding cosmological background both and vary in time. Initially the field is in
the broad resonance regime () for , but it gradually enters the narrow resonance regime
(). Since the field passes many instability and stability bands, the growth index
stochastically changes with the cosmic expansion. The non-adiabaticity of the change of the
frequency can be estimated by the quantity

where the non-adiabatic regime corresponds to . For small and we have
around , where are positive integers. This corresponds to the time at which the Ricci
scalar vanishes. Hence, each time crosses 0 during its oscillation, the non-adiabatic particle
production occurs most efficiently. The presence of the mass term tends to suppress the
non-adiabaticity parameter , but still it is possible to satisfy the condition around
.

For the model (3.6) it was shown in [591] that massless particles are resonantly amplified for
. Massive particles with of the order of can be created for . Note that in the
preheating scenario based on the model the parameter decreases
more rapidly () than that in the model (3.6) [354]. Hence, in our geometric preheating scenario,
we do not require very large initial values of [such as ] to lead to the efficient parametric
resonance.

While the above discussion is based on the linear analysis, non-linear effects (such as the mode-mode
coupling of perturbations) can be important at the late stage of preheating (see, e.g., [354, 342]). Also the
energy density of created particles affects the background cosmological dynamics, which works as a
backreaction to the Ricci scalar. The process of the subsequent perturbative reheating stage can be
affected by the explosive particle production during preheating. It will be of interest to take into
account all these effects and study how the thermalization is reached at the end of reheating.
This certainly requires the detailed numerical investigation of lattice simulations, as developed
in [255, 254].

At the end of this section we should mention a number of interesting works about gravitational
baryogenesis based on the interaction between the baryon number current
and the Ricci scalar ( is the cut-off scale characterizing the effective theory) [179, 376, 514]. This
interaction can give rise to an equilibrium baryon asymmetry which is observationally acceptable, even for
the gravitational Lagrangian with close to 1. It will be of interest to extend the analysis
to more general f (R) gravity models.